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Essential norms of composition operators between Bloch type spaces

Author: Ruhan Zhao
Journal: Proc. Amer. Math. Soc. 138 (2010), 2537-2546
MSC (2000): Primary 47B33; Secondary 46E15
Published electronically: February 26, 2010
MathSciNet review: 2607883
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Abstract: For $ \alpha>0$, the $ \alpha$-Bloch space is the space of all analytic functions $ f$ on the unit disk $ D$ satisfying

$\displaystyle \Vert f\Vert _{B^{\alpha}}=\sup_{z\in D}\vert f'(z)\vert(1-\vert z\vert^2)^{\alpha}<\infty. $

Let $ \varphi$ be an analytic self-map of $ D$. We show that for $ 0<\alpha,\beta<\infty$, the essential norm of the composition operator $ C_{\varphi}$ mapping from $ B^{\alpha}$ to $ B^{\beta}$ can be given by the following formula:

$\displaystyle \Vert C_{\varphi}\Vert _e=\left(\frac{e}{2\alpha}\right)^{\alpha}\limsup_{n\to\infty} n^{\alpha-1}\Vert\varphi^n\Vert _{B^{\beta}}. $

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Ruhan Zhao
Affiliation: Department of Mathematics, The College at Brockport, State University of New York, Brockport, New York 14420

Keywords: Composition operators, essential norms, Block type spaces
Received by editor(s): July 16, 2009
Received by editor(s) in revised form: October 25, 2009, November 2, 2009, and November 11, 2009
Published electronically: February 26, 2010
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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